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Modern Analysis 2
Test 3
Answer TWO questions.
1. Let (
f
n
)
∞
n
=0
be a sequence of measurable functions. Prove that each of the
following sets is measurable:
(e)
E
=
{
x
:
f
n
(
x
)
> e
n
}
;
(f)
F
=
{
x
:
f
n
(
x
)
<
0 for ﬁnitely many
n
}
;
(h)
H
=
{
x
:
n >
0
⇒
f
n
(
x
) =
1
2
(
f
n
+1
(
x
) +
f
n

1
(
x
))
}
;
(i)
I
=
{
x
: lim
n
f
n
(
x
)
∈
Z
}
.
2. Let
f
:
X
→
R
be measurable and
φ
: [0
,
∞
)
→
[0
,
∞
) be increasing.
Prove that if
t >
0 then
Z
(
φ
◦ 
f

)d
μ
>
φ
(
t
)
μ
{
f

>
t
}
and that if also
μ
(
X
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Unformatted text preview: ) < ∞ and φ 6 M then Z ( φ ◦  f  )d μ 6 φ ( t ) μ ( X ) + Mμ { f  > t } . 3. Say f n μ → f iﬀ ( ∀ ε > 0) μ { ff n  > ε } → . Let μ ( X ) < ∞ . Deﬁne d ( f,g ) = Z  fg  1 +  fg  d μ. Show that f n μ → f ⇔ d ( f,f n ) → . 1...
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This note was uploaded on 07/08/2011 for the course MAA 5229 taught by Professor Robinson during the Spring '11 term at University of Florida.
 Spring '11
 Robinson

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